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Adinkras and arithmetical graphs have divergent origins. In the spirit of Feynman diagrams, adinkras encode representations of supersymmetry algebras as graphs with additional structures. Arithmetical graphs, on the other hand, arise in algebraic geometry, and give an arithmetical structure to a graph. In this thesis, we will interpret adinkras as arithmetical graphs and see what can be learned. Our work consists of three main strands. First, we investigate arithmetical structures on the underlying graph of an adinkra in the specific case where the underlying graph is a hypercube. We classify all such arithmetical structures and compute some of the corresponding volumes and linear ranks. Second, we consider the case of a reduced arithmetical graph structure on the hypercube and explore the wealth of relationships that exist between its linear rank and several notions of genus that appear in the literature on graph theory and adinkras. Third, we study modifications of the definition of an arithmetical graph that incorporate some of the properties of an adinkra, such as the vertex height assignment or the edge dashing. To this end, we introduce the directed arithmetical graph and the dashed arithmetical graph. We then explore properties of these modifications in an attempt to see if our definitions make sense, answering questions such as whether the volume is still an integer and whether there are still only finitely many arithmetical structures on a given graph.

Nesta tese estudamos o modelo CP^(N-1) em (2+1) dimensões do espaco-tempo, onde o campo básico está na representacão fundamental. Diferentemente do caso em que o campo básico está na representacão adjunta, já estudado na literatura, o modelo por nós estudado se reduz ao modelo supersimétrico usual no limite comutativo. Analisamos a estrutura de fase e calculamos as correcões dominantes e subdominantes na expansão 1/N. Provamos que a teoria é livre de singularidades infravermelhas não integráveis e é renormalizável na ordem dominante. A funcão de vértice de dois pontos do campo básico é calculada e renormalizada de uma forma explicitamente supersimétrica na ordem subdominante. / In this thesis we sutudy the noncommutative supersymmetric CP^(N-1) model in (2+1) space-time dimensions, where the basic field is in the fundamental representation which, differently to the adjoint representation already studied in the literature, goes to the usual supersymmetric model in the commutative limit. We analyse the phase structure of the model and calculate the leading and subleading corrections in the 1/N expansion. We prove that the theory is free of non-integrable IR/UV infrared singularities and is renormalizable in the leading order. The two point vertex function of the basic field is also calculated and renormalized in an expliciitly supersymmetic way up to subleading order.

Diese Arbeit beschreibt die Suche nach dem Higgs Boson, welches in vielen theoretischen Modellen der Teilchenphysik vorhergesagt wird. Das Higgs Boson ist die Konsequenz der spontanen Symmetriebrechung, welche den Teilchen Masse verleiht. Zur Suche werden e+e- Annihilationen bei Schwerpunktenergien bis 209 GeV analysiert, welche vom Experiment L3 am Speicherring LEP in den Jahren 1998 bis 2000 registriert wurden. Die Suche erfolgte in allen relevanten Endzustaenden, wobei der Endzustand mit vier hadronischen Jets im Detail behandelt wird. Die Daten werden mit den Erwartungen eines Signals in verschiedenen Modellen bei Beruecksichtigung der bekannten Untergrundprozesse verglichen oder es wird modellunabhaengig nach der Erzeugung skalarer Teilchen gesucht. Die Produktion von Higgs Bosonen konnte nicht nachgewiesen werde. Die Daten wurden daher benutzt, um neue Grenzen fuer Parameter der Modelle oder der Kopplungen zu setzen. Als erstes wird die Suche nach dem Higgs Boson im Standard Modell der elektroschwachen Wechselwirkung beschrieben. Die Produktion des Higgs Bosons wird bei LEP Energien ueber die Higgs-Strahlung und der Zerfall des Higgs Bosons in ein Paar von b-Quarks vorhergesagt. Die Analysen beruhen daher wesentlich auf der Erkennung von B-Hadronen. Der HZ->qqqq Endzustand wird im Detail untersucht, und die Ergebnisse werden mit den anderen Kanaelen : HZ->qqvv, HZ->qql+l- und HZ->tau+tau- qq kombiniert. Die untere Massengrenze fuer das Higgs Boson wird zu mH > 112.0 GeV auf 95% Vertrauensniveau, bestimmt. Ausserdem werden Grenzen auf die HZZ Kopplung abgeleitet. Im minimalen supersymmetrischen Modell (MSSM) werden fuenf Higgs Bosonen vorhergesagt. Zur Higgs-Strahlung kommt die Paarproduktion von Higgs Bosonen, e+e- -> hA , hinzu. Die Ergebnisse der Suche im Standard Modell werden durch die Suche in den Endzustaenden bb tau+tau- (tau+tau- bb), bbbb und hZ->AAqq ergaenzt. Im Rahmen von drei Standard-Szenarien, benannt als "mh-max", "no mixing" und "large-mu" werden untere Grenzen auf die Higgs Boson Massen von mh > 84.5 GeV und mA > 86.3 GeV fuer tan(beta) > 0.7 abgeleitet. Weiterhin werden im "mh-max" Szenario 0.55 < tan(beta) < 2.2, im "no mixing" Szenario 0.4 < tan(beta) < 4.9, und im "large-mu" Szenario 0.7 < tan(beta) < 6.2 ausgeschlossen. Eine modellunabhaengige Suche nach dem Prozess e+e- -> hZ wird fuer den vier-Jet Endzustand durchgefuehrt. In Kombination mit den Ergebnissen von den anderen Kanaelen werden Grenzen fuer die hZZ Kopplung bestimmt. Wird die hZZ Kopplung auf den Wert im Standard Modell gesetzt und der Zerfall des Higgs Bosons zu 100% in Hadronen angenommen, ergibt sich als Grenze der Higgs Boson Masse, mh > 97 GeV. Modellunabhaengige obere Grenzen fuer die hAZ Kopplung werden aus der Suche nach der Paarerzeugung von Higgs Bosonen in den Kanaelen hA->qqqq und hA->qq tau+tau- in Abhaengigkeit von den Higgs Boson Massen abgeleitet. Die Resultate aus der Kombination aller LEP Experimente werden fuer die oben genannten Analysen vorgestellt. Die Perspektiven der Higgs Boson Suche an den TEVATRON und LHC Speicherringen werden diskutiert und die Higgs Boson Physik an kuenftigen e+e- Linearbeschleunigern behandelt. Ein Linearbeschleuniger wie TESLA waere ideal fuer Untersuchungen eines leichten Higgs Bosons. Fuer ein Higgs Boson mit einer Masse zwischen 120 und 180 GeV kann mH mit einer Praezision von 40 bis 70 MeV bestimmt werden. Im gleichen Massenbereich ist die Messung des Wirkungsquerschnitts, weitgehend modellunabhaengig, mit einem relativen Fehler von 2.6 bis 3.8% moeglich. In vier-Fermion und sechs-Fermion Endzustaenden werden topologische Wirkungsquerschnitte, definiert als Produkt des totalen Wirkungsquerschnitts fuer e+e- -> HZ mit dem Verzweigungsverhaeltnis eines Zerfallskanals, untersucht. Fuer den Messfehler werden 1.1% fuer HZ->bbqq und 13% fuer HZ->W+W-l+l- bei mH = 120 GeV abgeschaetzt. Die Gesamtheit dieser und weiterer Messungen erlaubt eine genaue Bestimmung des Higgs Boson Profils und gibt Aufschluss ueber die Struktur des Higgs Sektors in der Natur. / This thesis is devoted to the search for neutral Higgs bosons predicted by various theoretical models. The Higgs boson arises as a result of spontaneous breaking of SU(2) symmetry leading to the generation of masses of fermions and weak bosons. The search is done in all experimentally related channels using the data collected at center-of-mass energies up to 209 GeV in the years 1998-2000 with the L3 detector at the Large Electron Positron collider, LEP. Here the study of the final states characterised by four jets is described in detail. For other final states the analyses are briefly reviewed and the results are reported. The data are compared with the expectation from the Standard Model background processes and with various signal hypotheses. A model independent search for neutral Higgs bosons is also performed. No evidence for the production of Higgs bosons is found. New mass limits are determined superseding previous mass limits established by L3 and other experiments. First I describe the analysis searching for the Standard Model Higgs Boson. Its production at LEP is expected mainly via the Higgs-strahlung process. In the mass range accessible at LEP the Standard Model Higgs Boson is predicted to decay dominantly into a pair of b and anti-b quarks, hence the dedicated analyses are optimised for the H->bb decay mode. The four-jet signal topology is investigated and then combined with the other search channels leading to a lower mass limit of mH > 112.0 GeV at 95% C.L.. The results of the search are also interpreted in terms of limits on the HZZ coupling. In the Minimal Supersymmetric Standard Model (MSSM) the Higgs sector is extended to five physical states. The Higgs-strahlung process is complemented by the mechanism of the Higgs boson pair production e+e- -> hA. Therefore, for the interpretation of the results in the framework of the MSSM the Standard Model analyses are combined with the hA -> bb tau+tau-, hA -> bbbb and hZ->AAqq channels. Three benchmark MSSM scenarios denoted "mh-max", "no mixing" and "large-mu" are considered. Using L3 data the lower bounds on the Higgs boson masses mh > 84.5 GeV mA > 86.3 GeV are derived at 95% C.L. for tan(beta) > 0.7. For the "mh-max", "no mixing" and "large-mu" scenarios, ranges 0.55 < tan(beta) < 2.2, 0.4 < tan(beta) < 4.9 and 0.7 < tan(beta) < 6.2, respectively, are ruled out. A model independent search for the Higgs-strahlung process with subsequent decay of h into hadrons is carried out in the four-jet channel. The results of the analysis are then combined with the other channels. A limit on the hZZ coupling as a function of the Higgs boson mass is derived. The results of L3 combined search establish a 95% C.L. lower mass limit, mh > 97 GeV, for a hadronically decaying Higgs boson assuming the cross section of the Higgs-strahlung process to be equal to the value predicted by the Standard Model and the branching fraction of the Higgs boson into hadrons equal to 100%. Analyses are developed to search exclusively for the hA -> bbbb, hA -> qqqq, hA -> bb tau+tau- and hA -> qq tau+tau- final states. Results of these analyses are translated into a 95% C.L. upper limit on the hAZ coupling as a function of Higgs boson masses. Searches for neutral Higgs bosons carried out by the L3 collaboration are combined with searches performed in other LEP experiments. The results of this combination are reported. The perspectives of Higgs boson searches at TEVATRON and LHC are briefly reviewed. The prospects of Higgs physics at a future linear e+e- collider are discussed. The potential of the TESLA detector foreseen at the TESLA linear collider for the determination of Higgs boson properties is studied. The Higgs boson masses 120, 150 and 180 GeV are considered. It is shown that a precision of 40 - 70 MeV in the measurement of the Higgs boson mass can be achieved. A model independent method to measure the e+e- -> HZ cross section is proposed. The method is based on the study of the inclusive HZ -> X e+e- and X mu+mu- channels. The relative error in the determination of the cross section varies between 2.6% and 3.8% for Higgs boson mass ranging from 120 GeV to 180 GeV. For the four-fermion and six-fermion final states arising from the Higgs-strahlung process the accuracy of the measurement of a topological cross section, defined as the product of the Higgs-strahlung cross section and the branching fraction of the specific final state, is investigated. The relative uncertainty of this measurement varies from 1.1% and 13%, depending on final state and Higgs boson mass. These and other measurements will allow to determine the profile of the Higgs boson and give insight into the structure of the Higgs sector in nature.

In dieser Arbeit behandeln wir exakte, nicht-perturbative Ergebnisse, die mithilfe der superkonformen Index-Technik, in supersymmetrischen Eichtheorien mit vier Superladungen (d. h. N=1 Supersymmetrie in vier Dimensionen und N=2 in drei Dimensionen) gewonnen wurden. Wir benutzen die superkonforme Index-Technik um mehrere Dualitäts Vermutungen in supersymmetrischen Eichtheorien zu testen. Wir führen Tests der dreidimensionalen Spiegelsymmetrie und Seiberg ähnlicher Dualitäten durch. Das Ziel dieser Promotionsarbeit ist es moderne Fortschritte in nicht-perturbativen supersymmetrischen Eichtheorien und ihre Beziehung zu mathematischer Physik darzustellen. Im Speziellen diskutieren wir einige interessante Identitäten der Integrale, denen einfache und hypergeometrische Funktionen genügen und ihren Bezug zu supersymmetrischen Dualitäten in drei und vier Dimensionen. Methoden der exakten Berechnungen in supersymmertischen Eichtheorien sind auch auf integrierbare statistische Modelle anwendbar. Dies wird im letzten Kapitel der vorliegenden Arbeit behandelt. / In this thesis we discuss exact, non-perturbative results achieved using superconformal index technique in supersymmetric gauge theories with four supercharges (which is N = 1 supersymmetry in four dimensions and N = 2 supersymmetry in three). We use the superconformal index technique to test several duality conjectures for supersymmetric gauge theories. We perform tests of three-dimensional mirror symmetry and Seiberg-like dualities. The purpose of this thesis is to present recent progress in non-perturbative supersymmetric gauge theories in relation to mathematical physics. In particular, we discuss some interesting integral identities satisfied by basic and elliptic hypergeometric functions and their relation to supersymmetric dualities in three and four dimensions. Methods of exact computations in supersymmetric theories are also applicable to integrable statistical models, which we discuss in the last chapter of the thesis.

Cette thèse décrit quelques développements dans les techniques de calcul des amplitudes de diffusion en théorie supersymétrique de champ de jauge . L'accent est mis sur les relations de récurrence on-shell et sur l'utilisation de méthodes d'unitarité pour des calculs de boucle. En particulier, la récurrence on-shell est liée aux règles BCFW pour calculer les amplitudes de jauge au niveau des arbres. Les combinaisons de techniques de coupe d'unitarité et la récurrence sont utilisées pour calculer les amplitudes de boucle, et finalement, à partir des amplitudes, pour obtenir la fonction de corrélation énergie-énergie en théorie de super-Yang-Mills N = 4 à l'aide de la représentation de Mellin-Barnes. Dans le dernier chapitre, nous tentons de trouver un contour convergent pour les intégrales de Mellin Barnes en multi-dimension obtenu par une certaine approximation d'un contour de phase stationnaire.

There are hints coming from some scenarios of modern String and Quantum Gravity theories that Lorentz invariance may not be an exact symmetry of Nature. The study of Lorentz violating theories therefore provides an insight into ultraviolet physics. We employ the Effective Field Theory technique to study the most general extension of the Standard Model and its Supersymmetric modifications with Lorentz-violating interactions of mass dimension five. We provide a complete classification of the interactions in these theories and determine a typical experimental sensitivity to the size of Lorentz violation. A detailed study of the operators that induce CPT-odd Electric Dipole Moments is performed and it is shown that they yield an independent constraint on Lorentz violating physics. We provide an application of Lorentz violating physics to the problem of generation of baryon asymmetry of the universe. A scenario of Leptogenesis driven by CPT-odd interactions is considered and confronted with experimental constraints on Lorentz-violating physics.

The supersymmetric solutions of PT-/non-PT symmetric and Hermitian/non-Hermitian forms of quantum systems are obtained by solving the Schr&Auml / odinger equation with the deformed Morse, Hulth&para / en, P&Auml / oschl-Teller, Hyperbolic Kratzer-like, Screened Coulomb, and Exponential-Cosine Screened Coulomb (ECSC) potentials. The Hamiltonian hi- erarchy method is used to get the real energy eigenvalues and corresponding wave functions.

Nesta tese estudamos o modelo CP^(N-1) em (2+1) dimensões do espaco-tempo, onde o campo básico está na representacão fundamental. Diferentemente do caso em que o campo básico está na representacão adjunta, já estudado na literatura, o modelo por nós estudado se reduz ao modelo supersimétrico usual no limite comutativo. Analisamos a estrutura de fase e calculamos as correcões dominantes e subdominantes na expansão 1/N. Provamos que a teoria é livre de singularidades infravermelhas não integráveis e é renormalizável na ordem dominante. A funcão de vértice de dois pontos do campo básico é calculada e renormalizada de uma forma explicitamente supersimétrica na ordem subdominante. / In this thesis we sutudy the noncommutative supersymmetric CP^(N-1) model in (2+1) space-time dimensions, where the basic field is in the fundamental representation which, differently to the adjoint representation already studied in the literature, goes to the usual supersymmetric model in the commutative limit. We analyse the phase structure of the model and calculate the leading and subleading corrections in the 1/N expansion. We prove that the theory is free of non-integrable IR/UV infrared singularities and is renormalizable in the leading order. The two point vertex function of the basic field is also calculated and renormalized in an expliciitly supersymmetic way up to subleading order.

The Standard Model of particle physics attempts to unify the fundamental forces in the Universe (except gravity). Over the years it has been tested in numerous experiments. While these experimental results strengthen our understanding of the SM, they also point out directions for physics beyond the SM. In this thesis we assume supersymmetry (SUSY) to be the new physics beyond the SM. We have tried to analyze the present status of low energy SUSY after the recent results from direct (collider) and indirect (flavor, dark matter) searches .We have tried to see the complementarity between these apparently different experimental results and search strategies from the context of low energy SUSY. We show that such complementarity does exist in well-defined models of SUSY breaking like mSUGRA, NUHM etc. The first chapter outlines the present status of the SM and discusses about the unanswered questions in SM. Keeping SUSY as the new physics beyond the SM, we also detail about its present experimental status. Chapter1 ends with the motivation and comprehensive description about each chapter of the thesis. In chapter2, we present an introduction to formal structure of SUSY algebra and the structure of MSSM. One of the such complementarities we have studied is between flavor and dark matter. In general flavor violation effects are not considered when studying DM regions in minimal SUSY models like mSUGRA. If however flavor violation does get generated through non-minimal SUSY breaking sector, one of the most susceptible regions would be the co-annihilation region for neutralino DM. In chapter 3 we consider flavor violation in the sleptonic sector and study its implications on the stau co-annihilation region. In this work we have taken flavor violation between the right-handed smuon (˜µR) and stau (˜τR). Due to this flavor mixing the lightest slepton (ĺ1) is a flavor mixed state. We have studied the effect of such ĺ11’s in the ‘stau co-annihilation’ region of the parameter space, where the relic density of the neutralinos gets depleted due to efficient co-annihilation with the staus. Limits on the flavor violating insertion in the right-handed sleptonic sector mainly comes from BR(τ → µγ). These limits are weak in some regions of the Parameter space where cancellations happen with in the amplitudes. We look for overlaps in parameter space where both the co-annihilation condition as well as the cancellations with in the amplitudes occur. We have shown that in models with non-universal Higgs boundary conditions (NUHM) overlap between these two regions is possible. The effect of flavor violation is two fold: (a) It shifts the co-annihilation regions towards lighter neutralino masses and (b) the co-annihilation cross sections would be modified with the inclusion of flavor violating diagrams which can contribute significantly. In the overlap regions, the flavor violating cross sections become comparable and in some cases even dominant to the flavor conserving ones. A comparison among the different flavor conserving and flavor violating channels, which contribute to the neutralino annihilation cross-section, is presented. One of the challenges of addressing quantitatively the complementarity problems is the lack of proper spectrum generator (numerical tools which computes SUSY sparticle spectrum in the presence of flavor violation in the sfermionic sector). For the lack of a publicly available code which considers general flavor violating terms in the renormalization group equations (RGE) we have developed a SUSY spectrum calculator, named as SuSeFLAV .It is a code written in FORTRAN language and calculates SUSY particle spectrum (with in the context of gravity mediation) in type I seesaw, in the presence of heavy right handed neutrinos (RHN). SuSeFLAV also calculates the SUSY spectrum in other type of SUSY breaking mechanisms (e.g. gauge mediation). The renormalization group (RG) flow of soft-SUSY breaking terms will generate large off-diagonal terms in the slepton sector in the presence of this RHNs, which will give rise to sizable amount of flavor violating (LFV) decays at the weak scale. Hence, in this code we also calculate the different rare LFV decays like, µ → eγ, τ → µγ etc. In SuSeFLAV the user has the freedom to choose the scale of the RHNs as well as the mixing matrix in neutrino sector. It is also possible to choose the values of the SUSY breaking input parameters at the user defined scale. The details of this package is discussed in chapter 4. Many of the present studies of complementarity between the direct and indirect searches are inadequate to address realistic scenarios, where SUSY breaking could be much more general compared to the minimal models. The work in this thesis is a step to wards this direction. Having said that, in the present thesis we have considered modifications of popular models with either explicit flavor violating terms (in some sectors) or sources of flavor violation through new particles and new couplings motivated by strong phenomenological reasons like neutrino masses. It should be noted however, the numerical tool which has been developed during the thesis can be used to address more complicated problems like with complete flavor violation in models of SUSY breaking. One of the popular mechanisms of neutrino mass generation is the so called Seesaw Mechanism. Depending on the extra matter sector present in the theory there are three basic types of them. The type I seesaw, which has singlet bright-handed neutrinos, the type II seesaw contains scalar triplets and type III seesaw has additional fermionic triplets. One of the implications of the seesaw mechanism is flavor violation in the sfermionic sector even in the presence of flavor universal SUSY breaking. This leads to a complementarity between flavor experiments and direct SUSY searches at LHC. With the announcement of the results from the reactor neutrino oscillation experiments, the reactor mixing angle (θ13) in the neutrino mixing matrix (PMNS matrix) gets fixed to a rather large non-zero value. In SO (10) GUT theories neutrino Yukawa couplings of type I seesaw gets related to the up-type fermion sector of the SM. In chapter 5 we update the status of SUSY type I seesaw assuming SO (10)- like relations for neutrino Dirac Yukawa couplings and two cases of mixing, one large, PMNS-like, and another small, CKM-like, are considered. It is shown that for the large mixing case, only a small range of parameter space with moderate tan β is still allowed. It is shown that the renormalization group induced flavor violating slepton mass terms are highly sensitive to the Higgs boundary conditions. Depending on the choice of the parameters, they can either lead to strong enhancements or cancellations with in the flavor violating terms. We have shown that in NUHM scenario there could be possible cancellations which relaxes the severe constraints imposed by lepton flavor violation compared to mSUGRA. We further updated the flavor consequences for the type II seesaw in SUSY theories. As mentioned previously in type II seesaw neutrino mass gets generated due to exchange of heavy SU (2) L triplet Higgs field. The ratio of lepton flavor violating branching ratios (e.g. BR(τ → µγ) /BR (µ → eγ) etc.) are functions of low energy neutrino masses ans mixing angles. In chapter 6 we have analyzed how much these ratios become, after the experimental measurement of θ13, in the whole SUSY parameter space or in other words how much these ratios help to constrain the SUSY parameter space. We compute different factors which can affect this ratios. We have shown that the cMSSM-like scenarios, in which slepton masses are taken to be universal at the high scale, predict 3.5 BR(τ → µγ) / BR(µ → eγ) 30 for normal hierarchical neutrino masses. We Show that the current MEG limit puts severe constraints on the light sparticle spectrum in cMSSM-like model for seesaw scale with in1013 - 1015 GeV. These constraints can be relaxed and relatively light sparticle spectrum can be still allowed by MEG result in a class of models in which the soft mass of triplet scalar is taken to be non-universal at the GUT scale. In chapter 7 we have analyzed the effect of largen eutrino Yukawa couplings on the supersymmetric lightest Higgs mass. In July 2012, ATLAS and CMS collaboration have updated the Higgs search in LHC and found an evidence of a scalar particle having mass around 125 GeV. The one-loop contribution to Higgs mass mainly depends on the top trilinear couplings (At), the SUSY scale and the top Yukawa (Yt). Thus in models with extra large Yukawa couplings at the high scale like the seesaw mechanism ,the renormalization scaling of the At parameter can get significantly affected. This in turn can modify the light Higgs mass at the weak scale for the same set of SUSY parameters. We have shown in type I seesaw with (Yν ~ 3Yu) the light Higgs mass gets reduced by 2 - 3 GeV in most of the parameter rspace. In other words the SUSY scale must be pushed high enough to achieve similar Higgs mass compared to the cMSSM scenario. We have got similar effect in SUSY type III seesaw scenario with (Yν ~Yu) at the GUT scale. In chapter 8 we summarize the results of the thesis and discuss the possible future directions.